# double factorial

The *double factorial ^{}* of a positive integer $n$ is the product $n!!$ of the positive integers less than or equal to $n$ that have the same parity as $n$, that is,

$$n!!=n(n-2)(n-4)\mathrm{\cdots}{k}_{n}$$ |

where ${k}_{n}$ denotes $1$ if $n$ is an odd number^{} and $2$ if $n$ is an even number.

For example,

$$7!!=7\cdot 5\cdot 3\cdot 1=105$$ |

$$10!!=10\cdot 8\cdot 6\cdot 4\cdot 2=3840$$ |

Note that $n!!$ is not the same as $(n!)!$.

Observe that $(2n)!!={2}^{n}n!$ and $(2n+1)!!=\frac{(2n+1)!}{{2}^{n}n!}$.

Title | double factorial |
---|---|

Canonical name | DoubleFactorial |

Date of creation | 2013-03-22 12:24:54 |

Last modified on | 2013-03-22 12:24:54 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 9 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 05A10 |